I am not sure where you are based, but over in the UK the flu jab is given routinely every October/november time to everyone who has asthma, if i don't make an appointment to get it, the nurse pulls me in when i collect my monthly script for my meds! throwing up wnen i get a cold/flu is a regular occurrance for me, usually it is due to a build up of excessive mucus dripping down my throat during the night, too much for my tummy to process, so within an hour or so of waking up my stomach evacuates it!
having asthma makes you very suseptable to chest infections, you should not mess around waiting, the quicker they are cought the less long term damage is done. a 5 day course of strong antibiotics should halt the infection though it can take weeks for the gunk to clear. i am still coughing up gunk 4 weeks after my last one!

in the absence of going to the doctors, try putting :
1 cinnamon stick
2 inches of root ginger cut in strips
a good pinch of whole cloves (the more you put in and the longer you infuse for the stronger the numbing effect on your tongue and throat)
a teaspoon of runny honey
juice of 1 lemon

into a mug or flask, fill with boiling water, leave to infuse till acceptible temperature to drink, drink the liquid.
every ingredient has healing properties. when i ran a nursery school for 36 kids aged 2-5 years old i drank this like others drink tea or coffee, not once did i get even a sniffle let alone a cold or flu!

hope you feel better soon x take care x